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Theorem funcid 14022
Description: A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcid.b  |-  B  =  ( Base `  D
)
funcid.1  |-  .1.  =  ( Id `  D )
funcid.i  |-  I  =  ( Id `  E
)
funcid.f  |-  ( ph  ->  F ( D  Func  E ) G )
funcid.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
funcid  |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )

Proof of Theorem funcid
Dummy variables  m  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcid.x . 2  |-  ( ph  ->  X  e.  B )
2 funcid.f . . . . 5  |-  ( ph  ->  F ( D  Func  E ) G )
3 funcid.b . . . . . 6  |-  B  =  ( Base `  D
)
4 eqid 2404 . . . . . 6  |-  ( Base `  E )  =  (
Base `  E )
5 eqid 2404 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
6 eqid 2404 . . . . . 6  |-  (  Hom  `  E )  =  (  Hom  `  E )
7 funcid.1 . . . . . 6  |-  .1.  =  ( Id `  D )
8 funcid.i . . . . . 6  |-  I  =  ( Id `  E
)
9 eqid 2404 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
10 eqid 2404 . . . . . 6  |-  (comp `  E )  =  (comp `  E )
11 df-br 4173 . . . . . . . . 9  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
122, 11sylib 189 . . . . . . . 8  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
13 funcrcl 14015 . . . . . . . 8  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1412, 13syl 16 . . . . . . 7  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1514simpld 446 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
1614simprd 450 . . . . . 6  |-  ( ph  ->  E  e.  Cat )
173, 4, 5, 6, 7, 8, 9, 10, 15, 16isfunc 14016 . . . . 5  |-  ( ph  ->  ( F ( D 
Func  E ) G  <->  ( F : B --> ( Base `  E
)  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z ) ) (  Hom  `  E
) ( F `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  D ) `
 z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
 (  .1.  `  x ) )  =  ( I `  ( F `  x )
)  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) ) ) )
182, 17mpbid 202 . . . 4  |-  ( ph  ->  ( F : B --> ( Base `  E )  /\  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) (  Hom  `  E )
( F `  ( 2nd `  z ) ) )  ^m  ( (  Hom  `  D ) `  z ) )  /\  A. x  e.  B  ( ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) ) )
1918simp3d 971 . . 3  |-  ( ph  ->  A. x  e.  B  ( ( ( x G x ) `  (  .1.  `  x )
)  =  ( I `
 ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x
(  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D ) z ) ( ( x G z ) `  (
n ( <. x ,  y >. (comp `  D ) z ) m ) )  =  ( ( ( y G z ) `  n ) ( <.
( F `  x
) ,  ( F `
 y ) >.
(comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) )
20 simpl 444 . . . 4  |-  ( ( ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) )  ->  ( (
x G x ) `
 (  .1.  `  x ) )  =  ( I `  ( F `  x )
) )
2120ralimi 2741 . . 3  |-  ( A. x  e.  B  (
( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) )  ->  A. x  e.  B  ( (
x G x ) `
 (  .1.  `  x ) )  =  ( I `  ( F `  x )
) )
2219, 21syl 16 . 2  |-  ( ph  ->  A. x  e.  B  ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) ) )
23 id 20 . . . . . 6  |-  ( x  =  X  ->  x  =  X )
2423, 23oveq12d 6058 . . . . 5  |-  ( x  =  X  ->  (
x G x )  =  ( X G X ) )
25 fveq2 5687 . . . . 5  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
2624, 25fveq12d 5693 . . . 4  |-  ( x  =  X  ->  (
( x G x ) `  (  .1.  `  x ) )  =  ( ( X G X ) `  (  .1.  `  X ) ) )
27 fveq2 5687 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
2827fveq2d 5691 . . . 4  |-  ( x  =  X  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  X )
) )
2926, 28eqeq12d 2418 . . 3  |-  ( x  =  X  ->  (
( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  <->  ( ( X G X ) `  (  .1.  `  X )
)  =  ( I `
 ( F `  X ) ) ) )
3029rspcv 3008 . 2  |-  ( X  e.  B  ->  ( A. x  e.  B  ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  ->  ( ( X G X ) `  (  .1.  `  X )
)  =  ( I `
 ( F `  X ) ) ) )
311, 22, 30sylc 58 1  |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   <.cop 3777   class class class wbr 4172    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307    ^m cmap 6977   X_cixp 7022   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845    Func cfunc 14006
This theorem is referenced by:  funcsect  14024  funcoppc  14027  cofucl  14040  funcres  14048  fthsect  14077  catcisolem  14216  prfcl  14255  evlfcl  14274  curf1cl  14280  curfcl  14284  curfuncf  14290  yonedainv  14333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-ixp 7023  df-func 14010
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